Theorem lautcnv 38083

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautcnv	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Proof of Theorem lautcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautcnv.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 38078	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4		f1ocnv 6724	. . 3 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	3, 4	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6		eqid 2739	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
7	1, 6, 2	lautcnvle 38082	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
8	7	ralrimivva 3116	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
9	1, 6, 2	islaut 38076	. . 3 ⊢ (𝐾 ∈ 𝑉 → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
10	9	adantr 480	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
11	5, 8, 10	mpbir2and 709	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∀wral 3065   class class class wbr 5078  ^◡ccnv 5587  –1-1-onto→wf1o 6429  ‘cfv 6430  Basecbs 16893  lecple 16950  LAutclaut 37978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-map 8591  df-laut 37982

This theorem is referenced by:  ldilcnv  38108